## Graphical Representation of Asymmetric Graphoid Structures (2003)

Venue: | In Third International Symposium on Imprecise Probabilities and Their Applications |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Vantaggi03graphicalrepresentation,

author = {B. Vantaggi},

title = {Graphical Representation of Asymmetric Graphoid Structures},

booktitle = {In Third International Symposium on Imprecise Probabilities and Their Applications},

year = {2003},

pages = {560--574}

}

### OpenURL

### Abstract

Independence models induced by some uncertainty measures (e.g. conditional probability, possibility) do not obey the usual graphoid properties, since they do not satisfy the symmetry property. They are efficiently representable through directed acyclic l-graphs by using L-separation criterion.

### Citations

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Citation Context ...dependent of Y given Z”) induced by probability distributions has a long and rich tradition; one can distinguish three main classic approaches based on undirected graphs [12], directed acyclic graph=-=s [14]-=-, or chain graphs [15]. These graphical structure obey graphoid properties (symmetry, decomposition, weak union, contraction, intersection). On the other hand, the independence models based on the cla... |

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Citation Context ...dependence statements “X is independent of Y given Z”) induced by probability distributions has a long and rich tradition; one can distinguish three main classic approaches based on undirected gra=-=phs [12]-=-, directed acyclic graphs [14], or chain graphs [15]. These graphical structure obey graphoid properties (symmetry, decomposition, weak union, contraction, intersection). On the other hand, the indepe... |

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Citation Context ...tistical mechanics, physics, etc. [11]. The counter-intuitive situations cannot be avoided within the usual framework of conditional probability. In the more general framework (de Finetti [8], Dubins =-=[9]-=-), a definition of stochastic independence (called cs-independence), which avoids these critical situations, has been introduced in [4] and the main properties have been studied. We recall that the af... |

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Citation Context ...iterion for directed acyclic graphs has been recalled together with its main properties. This is very useful for effective description of independence models induced by different uncertainty measures =-=[1, 2, 4, 5, 6, 13, 16, 18, 19]-=-. In fact, these models cannot be represented efficiently by the well-known graphical models [12, 14], because the related separation criteria satisfy the symmetry property. In this paper, we have con... |

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Citation Context ... assessment is coherent if it is the restriction of a conditional probability defined on A × A 0 , where A is the algebra generated by {E1,H1,...,En,Hn}. A characterization of coherence was proven in=-= [3]-=-: Theorem 1 Let F be an arbitrary finite family of conditional events and C denote the set of atoms Cr generated by the events E1,H1,...,En,Hn. For a real function P on F the following two statements ... |

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Citation Context ...iterion for directed acyclic graphs has been recalled together with its main properties. This is very useful for effective description of independence models induced by different uncertainty measures =-=[1, 2, 4, 5, 6, 13, 16, 18, 19]-=-. In fact, these models cannot be represented efficiently by the well-known graphical models [12, 14], because the related separation criteria satisfy the symmetry property. In this paper, we have con... |

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Citation Context ...is [7] , statistical mechanics, physics, etc. [11]. The counter-intuitive situations cannot be avoided within the usual framework of conditional probability. In the more general framework (de Finetti =-=[8]-=-, Dubins [9]), a definition of stochastic independence (called cs-independence), which avoids these critical situations, has been introduced in [4] and the main properties have been studied. We recall... |

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Citation Context ...iterion for directed acyclic graphs has been recalled together with its main properties. This is very useful for effective description of independence models induced by different uncertainty measures =-=[1, 2, 4, 5, 6, 13, 16, 18, 19]-=-. In fact, these models cannot be represented efficiently by the well-known graphical models [12, 14], because the related separation criteria satisfy the symmetry property. In this paper, we have con... |

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Citation Context ...last stronger formulation in the more general framework of de Finetti [8]: Definition 1 Given a Boolean algebra A, a conditional probability on A × A 0 (with A 0 = A \ {/0}) is a function P(·|·) in=-=to [0,1], which sat-=-isfies the following conditions: (i) P(·|H) is a finitely additive probability on A for any H ∈ A 0 (ii) P(H|H) = 1 for every H ∈ A 0 (iii) P(E ∧ A|H) = P(E|H)P(A|E ∧ H), whenever E,A ∈ A a... |

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Citation Context ...ive, the independence model has a graphoid structure. Hence, the lack of intersection property is due to zero probability on some of the possible events. Actually, it is well-known (see, for example, =-=[4, 6]-=-) that the classic definition of stochastic independence presents counter-intuitive situations when zero or one probability events are involved: for example, a possible event with zero or one probabil... |

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Citation Context ...ive, the independence model has a graphoid structure. Hence, the lack of intersection property is due to zero probability on some of the possible events. Actually, it is well-known (see, for example, =-=[4, 6]-=-) that the classic definition of stochastic independence presents counter-intuitive situations when zero or one probability events are involved: for example, a possible event with zero or one probabil... |

4 |
The L-separation criterion for description of cs-independence models
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Citation Context ...rily symmetric, it follows that (U,W|S) l G �⇒ (W,U|S)l G . Actually, the lack of symmetry property depends on the notion of blocked path and not on the condition of logical separation (2). Theore=-=m 3 [17] Let G =-=- (V,E,B) be a graph. The following properties hold 1. (Decomposition property) (U,W ∪ Z|S) l G =⇒ (U,W|S)l G 2. (Reverse decomposition property) (U ∪ Z,W |S) l G =⇒ (U,W|S)l G 3. (Weak union p... |

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Citation Context |

2 |
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Citation Context ...e aforementioned definition agrees with the classic one when the probabilities of the relevant events are different from 0 and 1. The main properties connected with graphoid structures were proved in =-=[16]-=-: these independence models generally are not closed with respect to the symmetry property. Hence, the classic separation criterion are not apt to represent asymmetric independence statements, so in [... |

1 |
Comment on “Graphical models
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Citation Context ...robability values are interesting not only from a merely theoretical point of view, but they are met in many real problems, for example in medical diagnosis [7] , statistical mechanics, physics, etc. =-=[11]-=-. The counter-intuitive situations cannot be avoided within the usual framework of conditional probability. In the more general framework (de Finetti [8], Dubins [9]), a definition of stochastic indep... |